The problem is, I don't know how to research a wine's ingredients. Obviously, its not on the labels, and with so many brands and types on the market, how could I possibly check out the tyramine level in any one of them?

You can't research the ingredients, but you can research whether or not the wine has been put through primary and malolactic fermentation, each of which produce tyramines but I think the higher levels come from the secondary fermentation. I could be wrong, which is why you need to do some research.

In a decade or so, I expect these matters will be showing up on labels, as governments are beginning to show concern over potential allergens in fermented products.